Linkage Analysis

The primary goal of linkage analysis is to determine the location (chromosome and region) of genes influencing a specific trait. We accomplish this by looking for evidence of co-inheritance of the trait with other genes or markers whose locations are known, and locating genes close to one another.

Genes are markers that sit close together on a chromosome are called "linked" and are likely to be inherited together.
Genes on separate chromosomes are never linked
Genes that are far away from each other on a chromosome are likely to be separated during homologous recombination and are considered not linked

Recall Mendel's 2nd Law: The Principle of Independent Assortment - Alleles of a gene pair assort independently of other gene pairs. The segregation of one pair of alleles in no way alters the segregation of another pair of alleles EXCEPT when the genes are linked on a chromosome.

A haploid genotype (haplotype) is a combination of alleles at multiple loci that are transmitted together on the same chromosome. It contains 1) The alleles present at each locus (multi-locus genotype) and 2) which alleles are on the same chromosome.

When traits are linked it means they are always inherited together. Consider the example:

The above represents 2 different phenotytpes on a pea plant, Y is color and R is whether its round or wrinkled.

If the color and shape are unlinked then we would consider each box of a punnet square to have equal likelihood:

If the color and shape are completely linked then all Haploid gametes are RY or ry, that is to say R is always inherited with Y and r is always inherited with y:

Genes can also be partially linked so the likelihood of them being paired together more likely but not guaranteed.

Recombination

Recombinant don't inherit the same chromosomes as their parents, while non-recombinant have chromosomes inherited by their parents. Proportion of recombinants gives an indication of the distance along the chromosome between the two loci.

Consider Morgan's fruit fly experiment as an example:

Wing length and eye color are linked. So RN and rn should always be inherited together, but we can see the recombination Rn or rN in B1 that doesn't occur in parents.

The recombination fraction (theta) is the probability of recombination (i.e. the probability of an odd number of crossovers) between two loci.

The closer two loci are located on a chromosome the less likely they are to recombine
The further aprart they are, the more likely they are to recombine

𝜃 = 0 if:
- The genetic marker is the polymorphism causing the disease
- The marker is so close to the disease mutation that recombination can never occur
𝜃 = .5 if:
- There is a 50% chance that alleles at the two loci are inherited together. This happens when the two loci are
  - Very far apart on a chromosome
  - Located on two different chromosome

When loci are on different chromosomes, we observe recombination 50% of the time.

The line inbetween the allele means they are on the same chromosome.

Linkage Analysis

Linkage analysis estimates the genetic distance between genetic markers or between genetic markers and a trait locus with recombination. This allows us to know where a trait locus is in the genome when we know 1) the genetic distance between a disease locus and a marker AND 2) the location of the marker on the genome.

To perform a linkage analysis, estimate the recombination fraction between the disease locus and a marker locus and test:

H0: 𝜃 = 1/2 -> not linked i.e segregating independently

H1: 𝜃 < 1/2 -> linked, i.e. close together on the same chromosome

The most common test of linkage is Logarithm of the Odds (LOD):

LOD Score = log base 10 of the likelihood ratio = log₁₀(L(𝜃=𝜃₁)/L(𝜃=.5))

This is a transformation of the usual likelihood ratio test. The probability of observing R recombinants and N non-recombinants, where the recombinantion fraction is 𝜃, is as follows:

So the likelihood is:

We can ignore the constant because we willl be working with a ratio of likelihoods and the constant cancels.

When we know the exact number of recombinants and non-recombinants:

The MLE of 𝜃 is 𝜃_hat = R/(R+N) which is the value of 𝜃 that maximizes the LOD score and likelihood function.

LOD score > 3 -> linkage
LOD score (for a particular q value) < -2 -> no linkage
When the LOD scores is between 3 and -2, results are inconclusive. We might want to obtain additional individuals, additional families or utilize additional markers

Factors Influencing Linkage Analysis

Penetrance - the probability of expressing the disease given a specific genotype. Ex. reduced penetrance could be P(Disease | DD or Dd) < 1.0 for Autosomal Dominant

More markers

Overview of Linkage Analysis

Parametric linkage analysis assesses linkage between a marker and a locus

Need to specify a model for the inheritance of the disease
Need to specify risk allele frequency
Need to specify penetrances

Advantages

Most powerful approach when the model is correctly specified
It utilizes every family member's phenotypic and genotypic information
It provides a statistical test for linkage and for genetic heterogrneity

Disadvantages

Poor power if the genetic model is misspecified
Unaffected individuals may provide little information if penetrance is low
Can be difficult to recruit large families

Multipoint Linkage Analysis

Multipoint analysis incorporates multiple markers into the likelihood computation. It computes the likelihood that a disease is located at a certain position on a chromosome. The null hypothesis is that the disease locus is not on the chromosome. It has more information and thus is more powerful.

Affected Relative Pairs

Based on sets of affected pairs, no need to specify a model. Easier to collect but need a larger sample.

Alleles that are copies of the same allele from a common ancestor are called "identical-by-descent" or IBD for short.

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